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Transcript of Electromagnetic Waves Chapter 1. 2 Chapter Outlines Chapter 1 Electromagnetic Waves Faraday’s Law...
Electromagnetic Waves
Chapter 1
Chapter 1
2
Chapter OutlinesChapter OutlinesChapter 1 Electromagnetic Waves
Faraday’s Law Transformer and Motional EMFs Displacement Current Maxwell’s Equations Lossless TEM Waves EM Wave Fundamental and Equations EM Wave Propagation in Different Media EM Wave Reflection and Transmission at Normal or
Oblique Incidence
3
IntroductionIntroduction
In your previous experience in studying electromagnetic, you have learned about and experimented with electrostatics and magnetostatics … concentrating on static, or time invariant electromagnetic fields (EM Fields).
Henceforth, we shall examine situations where electric and magnetic fields are dynamic or time varying !!
4
Where :
• In static EM Fields, electric and magnetic fields
are independent each other, but in dynamic field
both are interdependent.
• Time varying EM Fields, represented by E(x,y,z,t)
and H(x,y,z,t) are of more practical value than
static EM Fields.
• In time varying fields, it usually due to
accelerated charges or time varying currents.
Introduction (Cont’d..)Introduction (Cont’d..)
5
In summary:
Stationary charges electrostatic fields
Steady currents magnetostatic fields
Time varying currents electromagnetic fields or waves
Introduction (Cont’d..)Introduction (Cont’d..)
1.1 Faraday’s Law1.1 Faraday’s Law
According to faraday’s experiment, a static magnetic field produces no current flow, but a time varying field produces an induced voltage called electromotive force or emf in a closed circuit, which causes a flow of current.
Faraday’s Law – the induced emf, Vemf in volts, in
any closed circuit is equal to the time rate of change of the magnetic flux linkage by the circuit.
Faraday’s Law (Cont’d..)Faraday’s Law (Cont’d..)
tVemf
Where,
The negative sign is a consequence of Len’z Law. If we consider a single loop, Faraday’s Law can be written as:
SB dtt
Vemf
An increasing magnetic field out of the page induces a current in (a) or an emf in (b). (c) The distributed resistance in a continuous conductive loop can be modeled as lumped resistor Rdist in series
with a perfectly conductive loop.
Faraday’s Law (Cont’d..)Faraday’s Law (Cont’d..)
Faraday’s Law (Cont’d..)Faraday’s Law (Cont’d..)
Generating emf requires a time varying magnetic flux linking the circuit. This occurs if the magnetic field changes with time ‘transformer emf’’ or if the surface containing the flux changes with time ‘motional emf’.The emf is measured around the closed path enclosing the area through which the flux is passing, can be written as:
SBLE dt
dVemf
It is clear that in time varying situation, both E and B are present and interrelated.
10
Example 1Example 1
Consider the rectangular loop moving with velocity
u=uyay in the field from an infinite length line current
on the z axis. Assume the loop has a distributed
resistance Rdist. Find an expression for the current in
the loop including its direction.
Solution to Example 1Solution to Example 1
First calculate the flux through the loop at an instant time,
aB2
0I aaa l
Where, la unit vector along the line current
a unit vector perpendicular from the line current to the field point
Remember ?
So,xyzl aaaaaa
xy
IIaaB
2200
Solution to Example 1 (Cont’d..)Solution to Example 1 (Cont’d..)
Arbitrarily choose dS in the +ax direction,
xdydzd aS So the flux can be easily calculated as:
yayIb
dzy
dyI
dydzy
Id
ay
y
b
lnln2
2
2
0
0
0
0
SB
Solution to Example 1 (Cont’d..)Solution to Example 1 (Cont’d..)
Then, we want to find how this flux changes with time,
yaydt
dIb
dt
dlnln
20
dt
dy
yay
Ib
dt
d
11
20
By chain rule,
By considering uy=dy/dt,
ayy
Iabu
dt
d y
20
Our emf is negative of this, where:
Solution to Example 1 (Cont’d..)Solution to Example 1 (Cont’d..)
ayy
IabuV
yemf
20
Since we considered dS in the +ax direction, our emf is
taken counterclockwise circulation. But since the emf is negative, our induced current is apparently going in the clockwise direction with value of:
dist
yind Rayy
IabuI
20
1.2 Transformer and motional emf1.2 Transformer and motional emf
The variation of flux with time as in previous equation maybe caused in three ways:
By having a stationary loop in a time varying B field.
(transformer emf)
By having a time varying loop area in a static B field.
(motional emf)
By having a time varying loop area in a time varying
B field.
Transformer and motional emf (Cont’d..)Transformer and motional emf (Cont’d..)
• Stationary Loop in Time Varying B Field
This is the case where a stationary conducting loop is in a time varying magnetic B field. The equation becomes:
SB
LE dt
dVemf
By applying Stokes Theorem in the middle term, we obtain:
SB
SE dt
dVemf
Transformer and motional emf (Cont’d..)Transformer and motional emf (Cont’d..)
This leads us to the point or differential form of Faraday’s Law,
t
B
E
Based on this equation, the time varying electric field is not conservative, or not equal to zero. The work done in taking a charge about a closed path in a time varying electric field, for example, is due to the energy from the time varying magnetic field.
Transformer and motional emf (Cont’d..)Transformer and motional emf (Cont’d..)
• Moving Loop in static B Field
When a conducting loop is moving in a static B field, an emf is induced in the loop. Recall that the force on a charge moving with uniform velocity in magnetic field,
BuQFm
BuQ
FE m
m
So then, we define the
motional electric field Em,
Transformer and motional emf (Cont’d..)Transformer and motional emf (Cont’d..)
If we consider a conducting loop moving with uniform velocity u as consisting of a large number of free electrons, the emf induced in the loop is:
L
memf ddV LBuLE
• Moving Loop in Time Varying Field
The total emf would be:
LS
memf ddt
dV LBuSB
LE
20
Example 2Example 2
The loop shown is inside a uniform magnetic field
B = 50 ax mWb/m2 . If side DC of the loop cuts
the flux lines at the frequency of 50Hz and the loop
lies in the yz plane at time t = 0, find the induced
emf at t = 1 ms.
Solution to Example 2Solution to Example 2
Since the B field is time invariant, the induced emf is motional, that is:
L
memf ddV LBuLE
Where,zDC dzdd aLL
aa
Lu
dt
d
dt
d loopmoving
1002,4 fcm
As u and dL is in cylindrical coordinates, transform B field into cylindrical coordinate (Chapter 1 in Electromagnetic Theory !! ):
Solution to Example 2 (Cont’d..)Solution to Example 2 (Cont’d..)
a-aaB sincos00 BB x
Where B0 = 0.05 , therefore:
z
z
B
BB
a
aaa
Bu
cos
0sincos
00 0
00
And
dz
dzdzBd
cos2.0
cos05.010004.0cos0
LBu
Solution to Example 2 (Cont’d..)Solution to Example 2 (Cont’d..)
mVdzVz
emf cos6cos2.003.0
0
To determine recall that,
Ctdtd
C is constant
at because the loop is in the yz plane! 2,0 t
Solution to Example 2 (Cont’d..)Solution to Example 2 (Cont’d..)
Hence,
2
t
Therefore,
mVt
tVemf
100sin6
2cos6cos6
So that at t = 1 ms,
mV.Vemf 8255)001.0(100sin6
1.3 Displacement Current1.3 Displacement Current
We recall from Ampere’s Circuital Law for static field,
cJH
EJ c
‘c’ subscript is used to identify it as a conduction current density, which related to electric field Ohm’s Law by:
But divergence of curl of a vector is identically zero,
JH 0
Displacement Current (Cont’d..)Displacement Current (Cont’d..)
The current continuity equation,
tc
vJ
dc JJH
We see that the static form of Ampere’s Law is clearly invalid for time varying fields since it violates the law of current continuity, and it was resolved by Maxwell introduction which what we called displacement current density,
Where Jd is the rate of
change of the electric flux density,
td
D
J
Displacement Current (Cont’d..)Displacement Current (Cont’d..)
The insertion of Jd was one of the major contribution
of Maxwell. Without Jd term, electromagnetic wave
propagation (e.g. radio or TV waves) would be
impossible. At low frequencies, Jd is usually
neglected compared with Jc. But at radio
frequencies, the two terms are comparable.
tc
D
JH
Therefore,
Displacement Current (Cont’d..)Displacement Current (Cont’d..)
By applying the divergence of curl, rearrange, integrate and apply Stoke’s Theorem, we can get the integral form of Ampere’s circuital Law:
dcc iidt
dd
SDSJLH
Do you really understand this displacement current??
Only formula and formula…????
Displacement Current (Cont’d..)Displacement Current (Cont’d..)
To have clear understanding of displacement current, consider the simple capacitor circuit of figure below.
A sinusoidal voltage source is applied to the capacitor, and from circuit theory we know the voltage is related to the current by the capacitance. i(t) here is the conduction current.
Displacement Current (Cont’d..)Displacement Current (Cont’d..)
Consider the loop surrounding the plane surface S1. By
static form of Ampere’s Law, the circulation of H must be equal to the current that cuts through the surface. But,
the same current must pass through S2 that passes
between the plates of capacitor.
Displacement Current (Cont’d..)Displacement Current (Cont’d..)
But, there is no conduction current passes through an ideal capacitor, (where J=0, due to σ=0 for an ideal
dielectric ) flows through S2. This is contradictory in view
of the fact that the same closed path as S1 is used.
But to resolve this conflict, the current passing through S2
must be entirely a displacement current, where it needs to be included in Ampere’s Circuital Law.
122 SSS
d dIt
Qd
tdd SJSDSJLH
So we obtain the same current for either surface though it is conduction current in S1 and displacement current in S2.
Displacement Current (Cont’d..)Displacement Current (Cont’d..)
The ratio of conduction current magnitude to the displacement current magnitude is called loss tangent, where it is used to measure the quality of the dielectric good dielectric will have very low loss tangent.
Other example for physical meaning:
Ji = current sourceJc= conducted current through resistorJd=displacement current through dielectric material
mi= magnetic current sourcemd=displacement magnetic current
1.4 Maxwell Equations1.4 Maxwell Equations
Below is the generalized forms of Maxwell Equations:
Maxwell Equations
Point or Differential
Form
Integral Form
Gauss’s Law
Gauss’s Law for Magnetic Field
Faraday’s Law
Ampere’s CircuitalLaw
v D
0 B
t
B
E
tc
D
JH
encQd SD
0 SB d
SBLE dt
d
SDSJLH dt
dd c
Maxwell Equations (Cont’d..)Maxwell Equations (Cont’d..)
It is worthwhile to mention other equations that go hand in hand with Maxwell’s equations.
BuEF q
EJ
HB
ED
tv
J
Lorent’z Force Equation
Constitutive Relations
Current Continuity Equation
Circuit - Field relations:Circuit - Field relations:
EJc GVVR
i 1
HB iL
ED
t
iLVL
dt
Hmd
VCqe
t
VCi c
c
dt
Ee
Field Relation Circuit Relation
Maxwell Equations (Cont’d..)Maxwell Equations (Cont’d..)
1.5 Lossless TEM Waves1.5 Lossless TEM Waves
Let’s use Maxwell’s equations to study the relationship between the electric and magnetic field components of an electromagnetic wave.
Consider an x-polarized wave propagating in the +z direction in some ideal medium characterized by µ and ε, with σ = 0.
Electromagnetic wave as x-polarized means that the E field vector is always pointing in the x or –x direction. Choose σ = 0 to make medium lossless for simplicity, as given by:
xztEtz aE cos, 0
Lossless TEM Waves (Cont’d..)Lossless TEM Waves (Cont’d..)
A plot of the equation E(z,0) = E0cos(z)ax at 10 MHz
in free space with E0 = 1 V/m.
Lossless TEM Waves (Cont’d..)Lossless TEM Waves (Cont’d..)
Upon application of Maxwell equations, we would also find the magnetic field propagates in the +z direction, but the field is always normal or perpendicular to the electric field vector the wave is said to propagate in a transverse electromagnetic wave mode, or TEM.
TEM Waves has no E field or H field components along the direction of propagation.We can apply Faraday’s Law, to
the propagating electric field equation previously.tt
HB
E
Solve the right hand side and the left hand side to get H !!
Lossless TEM Waves (Cont’d..)Lossless TEM Waves (Cont’d..)
yy
zyx
ztEztEz
ztEzyx
aa
aaa
E
sincos
00cos
00
0
dtztE
d
ztEt
y
y
aH
aH
sin
sin
0
0
Where,
CztE
y aH
cos0
We could see that the time varying E is the only source of H, if no conduction current given that can also generate H Thus C must be zero.
Plot of the equation
H(z,0) = (E0/) cos(–
z)ay at 10 MHz in free
space with
E0 = 1 V/m along with
the lighter plot of
E(z,0).
The amplitudes of E
and H are related by
Maxwell equations.
Lossless TEM Waves (Cont’d..)Lossless TEM Waves (Cont’d..)
Lossless TEM Waves (Cont’d..)Lossless TEM Waves (Cont’d..)
We can apply Ampere’s Circuital Law, to
the propagating magnetic field equation previously.tc
DJH
By considering lossless characteristic, taking the curl of H, equating the equation and then integrate it, we could get :
Try this!!!
pu
And propagation velocity relation:
1
puto get
-Solution-
42
Example 3Example 3
Suppose in free space that:
E(z,t) = 5.0 e-2zt ax V/m.
Is the wave lossless?
Find H(z,t).
Solution to Example 3Solution to Example 3
Since the wave has an attenuation term (e-2zt) it is clearly not lossless.
To find H,
2 2
2
5 10
5 0 0
x y z
zt zto y y
zt
e tex y z zte
a a aH
E a a
2 210 10, =zt zt
y yo o
td e dt te dt
H a H a
Therefore,
Solution to Example 3 (Cont’d..)Solution to Example 3 (Cont’d..)
udv uv vdu 2 and .ztu t dv e dt
This integral is solved by parts
where we let
We arrive at:
2 22
10 10
2 4zt zt
yo o
t Ae e
z z m
H a
1.6 EM Wave Fundamental and equations1.6 EM Wave Fundamental and equations
In free space, the constitutive parameters are σ = 0, µr = 1, εr = 1, so the Ampere’s Law and Faraday’s
Law equations become :
tt
H
E B
E 0
ttc
E
H D
JH 0
If there is some point in space a source of time varying E field, a H field is induced in the surrounding region. As this H field also changing with time, it in turn induces an E field. Energy is pass back and forth between E and H fields as they radiate away from the source at the speed of light.
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
The EM waves radiates spherically, but at a remote distance away from the source they resemble uniform plane wave. In a uniform plane wave, the E and H fields are orthogonal, or transverse to the direction of propagation ( to propagate in TEM mode ).
We will briefly review some of the fundamental features of waves before employing them in the study of electromagnetic. Consider time harmonic waves, represented by sine waves, rather than transient waves (pulses or step functions), generally as:
xz zteEtz aE cos, 0
The E field is a function of position (z) and time (t). It is always pointing in +x or –x direction x-polarized wave.
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
Where,
0
0
E
eE z
fT
12
f 2
Initial amplitude at z = 0
and exponential terms attenuation
Angular frequency
Phase constant
Phase shift
And important relations:
2
fdt
dzu p
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
E(0,t) = Exax = E0cos(t)ax. E(0,t) =Exax = E0cos(t + )ax.
E(z, 0) = E0cos(–z)ax. E(z, 0) = E0e–zcos (–z)ax.
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
Use Maxwell’s equations to derive formulas governing EM wave propagation.
Consider that the medium is free of any charge, where:
And linear, isotropic, homogeneous, and time invariant (simple media), whereby the Maxwell’s equation can be rewritten as :
0D
0H , 0E
HE ,
EEH
tt
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
Take curl of both sides of Faraday’s Law,
Consider position derivative acting on a time derivative in a homogeneous material,
Exchange the Faraday’s Law for the curl of H
Invoking a vector identity,
t
HE
HE
t
2
2
tttt
EEE
EE
AAA 2
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
So now we have :
2
22
tt
EE
EE
2
22
tt
EE
E
But, medium is charge free, so divergence of E is zero,
Helmholtz Wave Equation for E.
This can be broken up into three vector equations ( Ex, Ey and Ez)
53
Task 1 Task 1
By starting with Maxwell’s equations for
simple and charge free media, derive the
Helmholtz Wave Equation for magnetic
field, H
Solve this! Please submit the solution after class!
-Solution-
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
But, with interest on Helmholtz equations for time harmonic fields, the previous Helmholtz wave equation becomes :
ss jj EE 2s
s jt
EE
022 ss EE
because
Generally written in the form:
Where the propagation constant, γ, with real part is attenuation and an imaginary part is phase constant,
jjj )(
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
The value of and β in terms of the material’s constitutive parameters:
112
112
2
2
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
The Helmholtz wave equation for time harmonic magnetic fields, H :
022 ss HH
With loss of generality, we assume that x polarized traveling in the +az direction, where Es has only an x
component with function of z, since for plane wave that the fields do not vary in transverse direction, where in this case is xy plane. Then :
xxss zE aE )(
0)()()()( 2
2
2
2
2
2
2
zE
z
zE
y
zE
x
zExs
xsxsxs
022 zxsE-
By substituting this equation into the Helmholtz wave equation for E field, it becomes:
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
Hence,
With first and second term is zero :
0)(0)()( 2
2
22
2
2
zE
zzE
z
zExsxs
xs
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
This is a scalar wave equation, a linear homogeneous differential equation. A possible solution for this equation is :
If we let, then, zsx AeE zsxzsx Ae
z
EAe
z
E 22
2,
0,022 or
So, this equation, becomes:
Which has two solutions,
zsx
zsx
zsx
zsx
eEEAeE
eEEAeE
0
0
,,,0)2(
,,,0)1(
or
or
0)(22
2
zE
zxs
zzxs eEeEzE 00)(
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
The general solution is linear superposition of these two:
For the represents the E field amplitude of +z
traveling wave at z=0, by reinserting the vector, multiply
by time factor apply Euler’s identity and use the
propagation constant to get :
xz zteEtz aE cos, 0
tje
0E
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
For the represents the E field amplitude of negative z traveling wave at z=0, and similarly by previous approach to get :
0E
xz zteEtz aE cos, 0
The general instantaneous solution is superposition of these two solutions above.
xzz
s eEeE aE 00or generally as :
xz
xz zteEzteEtz aaE coscos, 00
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
Evaluating the curl of E to find:
yzz
s eEeE a-E 00
yzz
s ej
Ee
j
EaH
00
We can solve for H ,
If we assume that the wave propagates along +az and Hs has
only an y component, as what we had assume for Es yyss zH aH )(
The magnetic field can be found by applying Faraday’s Law :
ss j HE
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
We would have been led to the expression,
yzz
s eHeH aH 00
0H
0EBy making comparison, we can find a relationship between
and , or and
0E
0H
j
H
E
0
0intrinsic impedance (in ohms)
njn e
j
j
0
0
H
E 2tan
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
By plugged in the equation of intrinsic impedance into
equation of magnetic field, we could get:
ynz zte
Etz aH
cos, 0
Where E and H are out of phase by θn at any instant of time
due to the complex intrinsic impedance of the medium. Thus
at any time, E leads H or H lags E by θn . The ratio of
magnitude of conduction current density to displacement
current density in a lossy medium is:
tanjj s
s
d
c
E
E
J
Jloss tangent
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
Tan δ is used to determine how lossy a medium is, i.e.
• Good (lossless or perfect) dielectric if
• Good conductor if
,1tan
, 1tan
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
Basically, we can use Fleming’s Left Hand Rule to
determine the E, H and propagation direction ;
Propagation direction (first finger)
H direction (second finger)
E direction (thumb)
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
By knowing the EM wave’s direction of propagation,
given as unit vector ap, is the same as the cross
product of Es with unit vector, aE and Hs with unit
vector aH :
SPS
SPS
HaE
EaH
1
And also, a pair of simple formulas can be derived:
SS
SSP HE
HEa
HE
EH
HE
aaa
aaa
aaa
P
P
PIn addition to properties of cross product,
EM Wave Fundamental and equations (Cont’d..)EM Wave Fundamental and equations (Cont’d..)
Representation of waves; in (a), the wave travels in the ap
= +az direction and has Es = E0+ e–z ax and Hs = (E0
+/)e–z
ay. In (b), the wave travels in the ap = –az direction and has
Es = E0–ez ax along with Hs = –(E0
–/)ez ay.
68
Example 4 Example 4
Suppose in free space :
H(x,t) = 100 cos(2π x 107t – βx + π/4) az
mA/m. Find E(x,t).
Solution to Example 4 Solution to Example 4
0.100 , , 4
120 0.100 12
j x js z P x
j x j j x js P s x z y
e e
e e e e
H a a a
E a H a a a
We could find:
12 cos yt x E a
So then,
2 22 30 rad m
c f
7 212 cos 2 10
30 4 y
Vx t x
m
E a
Since free space is stated,
and then
70
Example 5 Example 5
Suppose:
E (x,y,t) = 5 cos(π x 106t – 3.0x + 2.0y) az
V/m.
Find :
H (x,y,t).
The direction of propagation, ap
Solution to Example 5 Solution to Example 5
We could find:
3 25 j x j ys ze eE a
Assume nonmagnetic material and therefore have:
3 2 3 210 15j x j y j x j ys s x yj j e e j e e E H a a
3 2 3 2 3 2 3 210 152.53 3.8j x j y j x j y j x j y j x j y
s x y x yo
j je e e e e e e e
j j
H a a a a
6 6 A( , , ) 2.53cos 10 3 2 3.80cos 10 3 2
mx yx y t x t x y x t x y H a a
So that,
Solution to Example 5 (Cont’d..) Solution to Example 5 (Cont’d..)
The direction of propagation :
s sP
s s
E H
aE H
6 4 6 419 12.65j x j y j x j ys s x ye e e e E H a a
Where,
And with the exponential terms canceling in the top
and bottom of the equation for ap, we have:
0.83 0.55 .P x y a a a
1.7 EM Wave Propagation In Different Media
1.7 EM Wave Propagation In Different Media
We will consider time harmonic field propagating in different types of media :-
Lossless, Charge-Free
Dielectrics
Conductors
• Lossless, Charge - Free
Charge free, ρv=0, medium has zero conductivity, σ=0.
This is the case where waves traveling in vacuum or free space (free of any charges). Perfect dielectric is also considered as lossless media.
EM Wave Propagation In Different Media (Cont’d..)EM Wave Propagation In Different Media (Cont’d..)
jjj )(
jjjjj 22)0(
, 0
Evaluate the propagation constant,
So,
Where,
Since , the signal does not attenuate as it travels lossless medium.
0
The propagation velocity,
1pu
EM Wave Propagation In Different Media (Cont’d..)EM Wave Propagation In Different Media (Cont’d..)
Evaluate the intrinsic impedance,
j
j
0
For lossless materials, E and H are always in phase. Again,
00
0
r
r
r
r 1200
intrinsic impedance of free space
76
In a lossless, nonmagnetic material with :
εr = 16, and H = 100 cos(ωt – 10y) az mA/m.
Determine : The propagation velocity The angular frequency The instantaneous expression for the
electric field intensity.
Example 6 Example 6
Solution to Example 6 Solution to Example 6
883 10
0.75 1016
p
r
c x mu x
s
The propagation velocity:
The angular frequency:
8 80.75 10 10 7.5 10p
radu x x
s
From given H field :
8( , ) 100cos 7.5 10 10 z
mAy t x t y
m H a
So, the time harmonic H field is:
Solution to Example 6 (Cont’d..)Solution to Example 6 (Cont’d..)
0.100 ,
1200.100 3
j ys z
j y j ys P s y z x
r
e
e e
H a
E a H a a aWhere,
8( , ) 9.4cos 7.5 10 10 x
Vy t x t y
m E a
Finally, the instantaneous expression for E field is:
• Dielectric
Treating a dielectric as lossless is often a good approximation, but all dielectrics are to some degree lossy finite conductivity, polarization loss etc. With finite conductivity, the E field gives rise to conduction current density results in power dissipation.
Thus, it will give a complex permittivity, complex propagation constant with attenuation constant greater than zero. The intrinsic impedance is also complex, resulting a phase difference between E and H fields.
EM Wave Propagation In Different Media (Cont’d..)EM Wave Propagation In Different Media (Cont’d..)
EM Wave Propagation In Different Media (Cont’d..)EM Wave Propagation In Different Media (Cont’d..)
• Conductor
In any decent conductor, the loss tangent, σ/ωε>>1 or σ>>ωε so that σ ≈ ∞, so that:
11
2,
2
f2
Therefore,
where interior
brackets becomes:
EM Wave Propagation In Different Media (Cont’d..)EM Wave Propagation In Different Media (Cont’d..)
The intrinsic impedance approximated by:
j
j
j
2
1 jj
045
2)1(
2jej
2
pu
f2
By considering leading to:
and also..
E leads H by 450
EM Wave Propagation In Different Media (Cont’d..)EM Wave Propagation In Different Media (Cont’d..)
A wave from air (free space) penetrates rapidly in a good conductor, with wavelength clearly much shorter.
In a good conductor, the large attenuation means the
penetration depth can be quite small, confining the fields
near the surface or skin, of the conductor skin depth.
A large attenuation means the fields cannot penetrate far into the conductor.
f
11
83
Example 7 Example 7
In a nonmagnetic material,
E(z,t) = 10 e-200z cos(2π x 109t - 200z) ax mV/m.
Find H(z,t)
Solution to Example 7 Solution to Example 7
Since = β, the media is a good metal, with µr =
1 we have:
22
9 7
200, or 10.13
1 10 4 10o
o
Sf
f mx x
45 452 28j je e
We could also find the intrinsic impedance,
Solution to Example 7 (Cont’d)Solution to Example 7 (Cont’d)
So, to calculate H,
1 1 1010 , 10z j z z j z z j z
s x s P s z x ye e e e e e
E a H a E a a a
The instantaneous expression for the magnetic field intensity.
200 9( , ) 360 cos 2 10 200 45zy
mAz t e x t z
m H a
1.8 EM Wave Reflection & Transmission at Normal Incidence1.8 EM Wave Reflection & Transmission at Normal Incidence
What happens when a EM wave is incident on a different medium? E.g. Light wave incident with mirror, most of it gets reflected but a portion gets transmitted (rapidly attenuating in the silver backing of the mirror.Consider a plane wave that are normally incident which means the planar boundary separating the two media is perpendicular to the wave’s propagation direction.
Generally, consider a time harmonic x-polarized electric field incident from medium 1 (µr1, εr1, σr1) to
medium 2 (µr2, εr2, σr2)
EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)
xzii zteEtz aE 10 cos),( 1
With incident field:
EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)
We have the following sets of equations:
yzjz
tts
xzjztt
s
yzjz
rrs
xzjzrr
s
yzjz
iis
xzjzii
s
eeE
eeE
eeE
eeE
eeE
eeE
aH
aE
aH
aE
aH
aE
22
22
11
11
11
11
2
0
0
1
0
0
1
0
0
Incident Fields
Reflected Fields
Transmitted Fields
tri EEE 000 ,, The E field intensities at z=0
The boundary conditions:
212121
21
tt
tt
HHKHHa
EE
i
riir
E
EEEE
0
0
12
1200
12
120 ,
Applying these boundary conditions at z=0 to get:
i
tiit
E
EEEE
0
0
12
200
12
20
2,
2
Reflection CoefficientTransmissio
n Coefficient
EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)
Try this!!
EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)
By comparison, 1
Consider a special case when medium 1 is a perfect dielectric (lossless,σ1=0) and medium 2 is a perfect
conductor (σ2= ∞). For this case, showing that the wave is totally reflected fields in perfect conductor
must vanish, so there can be no transmitted wave, E2 = 0.
0 ,1 ,02
The totally reflected wave combines with the incident wave to form a standing wave it stands and does not
travel, it consists of two traveling waves Ei and Er of
equal amplitudes but in opposite directions.
EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)EM Wave Reflection & Transmission at Normal Incidence (Cont’d..)
Standing wave pattern for an incident wave in a lossless medium reflecting off a second medium at z=0 where = 0.5.
1
1
min
max
E
ESWR
92
Example 8 Example 8
A uniform planar waves is normally incident
from media 1 (z < 0, σ = 0, µr = 1.0, εr = 4.0)
to media 2 (z > 0, σ = 0, µr = 8.0, εr = 2.0).
Calculate the reflection and transmission
coefficients seen by this wave.
Solution to Example 8 Solution to Example 8
The reflection coefficient ;
2 11 2
2 1
120 8; 60 , 120 240
24
This leads to:
240 60 30.60
240 60 5
and the transmission coefficient,
1 1.60
94
Example 9 Example 9
Suppose media 1 (z < 0) is air and media 2 (z
> 0) has εr = 16. The transmitted magnetic
field intensity is known to be:
Ht = 12 cos (ωt - β2z) ay mA/m.
Determine the instantaneous value of the
incident electric field.
Solution to Example 9 Solution to Example 9
We know that,
2 2
2
12t
j z j zt os y y
EmA mAe e
m m
H a a
From transmitted H field, we could find the
transmitted E field,
2t t2 o s
2
30 , so 12 , E 0.36 , and 1.13t
j zox
E mA V Ve
m m m
E a
2 1
2 1
3 21 ; , 1
5 5t i io o oE E E
Since we know the relation between transmitted E
field and incident E field,
12.83, so 2.83t
j zi ioo s x
EE e
E a
Thus, 1( , ) 2.83cos .x
Vz t t z
m E a
Solution to Example 9 (Cont’d..)Solution to Example 9 (Cont’d..)
1.9 EM Wave Reflection & Transmission at Oblique Incidence1.9 EM Wave Reflection & Transmission at Oblique Incidence
A uniform plane waves traveling in the ai
direction is obliquely incident from medium 1 onto medium 2.
Plane of incidence plane containing both a normal to the boundary and the incident’s wave propagation.In figure, the propagation direction is ai and the normal is az,
so the plane incidence is the x z plane. The angle of incidence, reflection and transmission is the angle that makes the field a normal to the boundary.
EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
When EM Wave in plane wave form obliquely incident on the boundary, it can be decomposed into:
Perpendicular Polarization, or transverse electric (TE) polarization The E Field is perpendicular or transverse to the plane of incidence.
Parallel Polarization, or transverse magnetic (TM) polarization The E Field is parallel to the plane of incidence, but the H Field is transverse.
We need to decompose into its TE and TM components separately, and once the reflected an the transmitted fields for each polarization determined, it can be recombined for final answer.
EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
• TE Polarization
EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
For TE polarization, the fields are summarized as follows:
ztxtzxj
tts
yzxjtt
s
zrxrzxj
rrs
yzxjrr
s
zixizxj
iis
yzxjii
s
tt
tt
rr
rr
ii
ii
eE
eE
eE
eE
eE
eE
aaH
aE
aaH
aE
aaH
aE
sincos
sincos
sincos
cossin
2
0
cossin0
cossin
1
0
cossin0
cossin
1
0
cossin0
2
2
1
1
1
1
Incident Fields
Reflected Fields
Transmitted Fields
EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
By applying Snell’s Law,
ri i
t
sin
sin
2
1
iTE
i
ti
tir EEE 0012
120 coscos
coscos
We could get:
iTE
i
it
it EEE 0021
20 coscos
cos2
TETE 1
Try to solve this!!!
• TM Polarization
EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
By similar geometric arguments, TM fields are:
yzxj
tts
ztxtzxjtt
s
yzxj
rrs
zrxrzxjrr
s
yzxj
iis
zixizxjii
s
tt
tt
rr
rr
ii
ii
eE
eE
eE
eE
eE
eE
aH
aaE
aH
aaE
aH
aaE
cossin
2
0
cossin0
cossin
1
0
cossin0
cossin
1
0
cossin0
2
2
1
1
1
1
sincos
sincos
sincos
Incident Fields
Reflected Fields
Transmitted Fields
EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
By applying Snell’s Law and employing the boundary conditions, we could get the following expressions relating the field amplitudes:
iTM
i
ti
it EEE 0021
20 coscos
cos2
iTM
i
it
itr EEE 0012
120 coscos
coscos
t
iTMTM
cos
cos1
EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)EM Wave Reflection & Transmission at Oblique Incidence (Cont’d..)
For TM polarizations, there exists an incidence angle
at which all of the wave is transmitted into the second
medium Brewster Angle, θi = θBA , where:
22
21
21
22
21
22
22 )(
sin
BA
2
11
1sin
r
rBA
When a randomly polarized wave such as light is incident on a material at the Brewster angle, the TM polarized portion is totally transmitted but a TE component is partially reflected.
106
Example 10 Example 10
A 100 MHz TE polarized wave with amplitude 1.0 V/m is obliquely incident from air (z < 0) onto a slab of lossless,
nonmagnetic material with εr = 25 (z > 0).
The angle of incidence is 40. Calculate:(a) the angle of transmission, (b) the reflection and transmission
coefficients,(c) the incident, reflected and transmitted for
E fields.
Solution to Example 10 Solution to Example 10
(a) 6
1 28
2 100 102.09 , 10.45 .
3 10r
x rad rad
c x m c m
1
2 2
1 1 1; sin sin 40 ; 7.4
5 5t tr
(b) 1 2
120120 ; 24
25
2 1
2 1
cos cos0.732; 1 0.268
cos cosi t
TE TE TEi t
Solution to Example 10 (Cont’d)Solution to Example 10 (Cont’d)
(c) For incident field:
2.09 sin 40 cos40 1.34 1.601 1j x zi j x j z
s y y
Ve e e
m
E a a
( , ) 1cos 1.34 1.60iy
Vz t t x z
m E a
Thus,
For reflected field:
0.732r io TE oE E
1.34 1.600.732r j x j zs y
Ve e
m E a
Leading to:
Thus,
( , ) 0.732cos 1.34 1.60ry
Vz t t x z
m E a
Solution to Example 10 (Cont’d)Solution to Example 10 (Cont’d)
Finally for transmitted field:
0.268t io TE oE E
2 sin cos 1.35 10.40.268 0.268t tj x zt j x j zs y y
Ve e e
m E a a
To get:
Therefore,
m
Vzxttz y
r aE 4.1035.1cos268.0),(
Solution to Example 10 (Cont’d)Solution to Example 10 (Cont’d)
Electromagnetic Waves
Electromagnetic Waves
End